This chapter discusses a computational verification method of existence of solutions for nonlinear elliptic equations. It proposes a numerical method for automatic proof of the existence of weak solutions for certain linear elliptic boundary value problems by computer. And its extension to the more general linear case is described. The main techniques in these works consist of the verification method by computer for the existential condition of solutions based on the infinite dimensional fixed point theorems. This chapter uses the properties of the solution for Poisson's equation and the results of error estimates for the finite element approximation as well as the method of interval arithmetic. This chapter formulates a numerical verification method which can be applicable to nonlinear elliptic boundary value problems.
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